If it were me I would max out the cards you can get on Chase until you get to 5/5 and then get the AA cards. In the Chase arena I would focus on Ultimate Rewards. They are much more flexible than AA or United miles and as good if not a better deal if you are flying coach since you can use any airline, any time and even earn FF miles on the ticket purchased with Ultimate Rewards. If you can, get the Sapphire Reserved card and one or two of the Freedom cards and transfer the miles earned on your Freedom Cards to maximize the value of your Ultimate Rewards (the Reserved cards allows you 50% more value).

After you have maxed out Chase, yes get the AA and the Barclays credit card. You could also get he Chase Marriott/Bonvoy card and then transfer the Bonvoy miles earned to most of the major airlines (at a much reduced rate).

Don’t discount getting the business cards as well.

Keep a spreadsheet of when you opened the cards, closed them and got the bonus. You will need those dates if you want to apply for the same cards at some point in the future.